now that I've changed my passwords on my gmail accounts, how do I change them in my Apple mail application?

Two things you can do:

1) Open Keychain Access, locate the relevant password, and change it there. Click the "Show password" checkbox, authenticate with your mac user account password as requesteed, and then type the new password and click "Save Changes".

2) In Mail itself, try to check mail for the relevant accounts. You will get an error that the password is invalid or similar. Type the new password as requested.

DaisyLady wrote:

1 - now that I've changed my passwords on my gmail accounts, how do I change them in my Apple mail application?

1 - Was this done by going into the Web Portal of Google G-mail ?

2 - Do not use G-Mail so perhaps going System Setting >> Internet Accounts and click on the G-mail Account(s) and change it there

3 -Or if using the IMAP Protocol on there Google e-mail accounts - remove the account and add them back using the New Passwords.

Some consideration specific the Google Application Specific Passwords usage

https://support.google.com/mail/answer/185833?hl=en-GB